Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Thomas A. Kaemming

Innovative Scientific Solutions, Inc., Dayton, Ohio 

Foundational Performance Analyses of Pressure 

Gain Combustion Thermodynamic 

Benefits for Gas Turbines

NASA/TM—2012-217443

May 2012

AIAA–2012–770
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Daniel E. Paxson

Glenn Research Center, Cleveland, Ohio

Thomas A. Kaemming

Innovative Scientific Solutions, Inc., Dayton, Ohio 

Foundational Performance Analyses of Pressure 

Gain Combustion Thermodynamic 

Benefits for Gas Turbines

NASA/TM—2012-217443

May 2012

AIAA–2012–770

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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subject to revision as analysis proceeds.
NASA/TM—2012-217443 1

Foundational Performance Analyses of Pressure Gain 

Combustion Thermodynamic Benefits for Gas Turbines

Daniel E. Paxson

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Thomas A. Kaemming

Innovative Scientific Solutions, Inc.

Dayton, Ohio 45440

Abstract

A methodology is described whereby the work extracted by

a turbine exposed to the fundamentally nonuniform flowfield 

from a representative pressure gain combustor (PGC) may be 

assessed. The method uses an idealized constant volume cycle, 

often referred to as an Atkinson or Humphrey cycle, to model 

the PGC. Output from this model is used as input to a scalable 

turbine efficiency function (i.e., a map), which in turn allows 

for the calculation of useful work throughout the cycle.

Integration over the entire cycle yields mass-averaged work 

extraction. The unsteady turbine work extraction is compared 

to steady work extraction calculations based on various 

averaging techniques for characterizing the combustor exit 

pressure and temperature. It is found that averages associated 

with momentum flux (as opposed to entropy or kinetic energy) 

provide the best match. This result suggests that momentumbased averaging is the most appropriate figure-of-merit to use 

as a PGC performance metric. Using the mass-averaged work 

extraction methodology, it is also found that the design turbine 

pressure ratio for maximum work extraction is significantly

higher than that for a turbine fed by a constant pressure 

combustor with similar inlet conditions and equivalence ratio.

Limited results are presented whereby the constant volume

cycle is replaced by output from a detonation-based PGC 

simulation. The results in terms of averaging techniques and 

design pressure ratio are similar.

1.0 Introduction

In the last decade, there has been renewed interest in the 

potential benefits of integrating pressure gain combustion 

(PGC) devices into gas turbine engines (Refs. 1 to 8).

Conventional gas turbine engines undergo steady, subsonic 

combustion which, though often described as constant 

pressure, actually results in a total pressure loss. PGC utilizes

any number of physical phenomena, including resonant pulsed 

combustion, constant volume combustion, or detonation, to 

affect a rise in cycle averaged, or effective pressure across the 

combustor, while consuming the same amount of fuel as the 

constant pressure combustor. In most implementations, the 

rise is achieved with few or no moving parts and without the 

need for positive displacement components (e.g., pistons). As 

applied to the gas turbine engine, PGC yields a theoretical 

thermodynamic benefit in the form of increased availability 

for the downstream work extracting turbine.

However, all practical PGC processes are fundamentally 

unsteady (though periodic). This presents two major 

challenges to PGC benefits analyses. First, the turbine engine 

community uses steady-state methods for computing 

performance. Therefore, most initial benefits assessments for 

advanced gas turbine components, including the unsteady 

PGC cycles, are done on a steady-state basis. Thus, some sort 

of averaging procedure is needed to transform the time 

varying PGC exhaust flow into a single state that adequately 

represents its availability (i.e., an equivalent total pressure and 

temperature). There are many ways to average a nonuniform 

flowfield, and each results in a different state. The first 

challenge then is to determine which method yields a state 

which, when used as input to a turbine performance map 

comes closest to predicting the actual work extraction.

The second challenge, closely related to the first, is to 

reasonably estimate ‘actual work extraction’. Most turbines 

are designed for steady flows and their performance is 

described on this basis. A methodology is therefore needed to 

determine the work extracted in the face of continually 

varying inlet conditions.

This paper preliminarily addresses these two challenges

using a fundamental approach. The PGC device is modeled 

using an idealized lumped volume (a.k.a., thermodynamic 

cycle) representation, and a calorically perfect gas assumption

(Refs. 9 and 10). Notionally, this may be envisioned as 

modeling just one combustor ‘can’ among the many which 

would comprise the complete combustor, with all undergoing

the same cyclic process. As will be shown, the lumped volume 

approach is convenient as it can be implemented algebraically, 

using purely thermodynamic relationships. This in turn 

provides rapid solutions for numerous parametric variations 

(e.g., combustor inlet conditions, equivalence ratios, turbine 

exit conditions, etc.).

The output of this model is used as input to a scalable 

turbine efficiency map. The turbine is assumed to operate in a 

quasi-steady fashion (i.e., aerodynamic response time is much 

less than the combustor cycle period). Each infinitesimal unit 

of mass leaving the combustor (at a different state) is passed 

through the map, providing an efficiency which, together with 
NASA/TM—2012-217443 2

the combustor state and the imposed turbine exit pressure 

allows the computation of an infinitesimal work increment.

Turbine exit pressure is assumed to be steady, consistent with 

the model assumption that there are multiple combustor 

‘cans.’ Integrating these increments over all the mass leaving 

the combustor provides the cycle mass-averaged work 

extracted.

Output from the combustor is also averaged using different 

methodologies (Refs. 11 to 13). The available total pressures 

and temperatures resulting from the different averaging 

methodologies are used in a standard, steady cycle analysis.

Using the same scalable turbine efficiency map, the work 

extraction based on the averaging methods is calculated. This 

work is compared to the cycle mass-averaged work over a 

broad range of operating parameters to statistically determine 

the accuracy, and therefore utility, of each averaging method

as a meaningful figure-of-merit with which to assess PGC

performance.

Details of the model, the turbine map, the averaging 

methods, and the statistical study are presented. Limited 

results are also presented in which the lumped volume PGC 

model output is replaced by output from a computational fluid 

dynamic simulation of a detonative PGC device operating at 

the design point. The intention is examine whether the PGC 

mode (e.g., constant volume or detonation) influences the 

findings.

2.0 Methodology

2.1 PGC Model Description

An idealized lumped volume (i.e., thermodynamic cycle) 

blowdown model is used to describe the simplified PGC 

process. The process may be imagined as follows. A vessel is 

initially filled with a known quiescent, combustible mixture, at 

the compressor exit total conditions (station 3). The mixture 

undergoes an instantaneous constant volume reaction to a 

post-combustion state. The resulting high pressure fluid is then 

expanded isentropically to a specified exit pressure through an 

ideal combustor exit/turbine entrance nozzle. When the 

pressure in the vessel reaches the initial pressure, the fill 

process begins. The throat of the exhaust nozzle is sized such 

that filling occurs very slowly (i.e., infinitesimally small fill 

Mach number). As such, filling occurs at the initial pressure, 

and the hot gases still in the tube continue to exit ideally 

through the exhaust nozzle. The vessel experiences no further 

expansion during the fill stage. There is no spatial variation.

The calorically perfect gas version of this process is 

mathematically described as follows. Beginning with the 

quiescent state, a nondimensional heat addition parameter may 

be defined.

R T ( a f )

h

q g t

v

γ +

∆ ′ ≡ 3 1 0 (1)

Here, and throughout the paper nondimensionalized 

quantities are denoted with the ′ superscript. Unless otherwise 

specified, they are nondimensionalized by the station 3 total 

conditions.

The post-combustion state is obtained through a constant 

volume heat addition process which takes the form:

( )

1

;

1 1 ;

3

3

0

3

= ρ

ρ ρ′ ≡

′ ≡ = ′

′ ≡ = + γ γ − ′

t

CV CV

CV

t

CV CV

t

CV CV

T

P

P P

q T

T T

(2)

Due to the assumption that the flow is isentropic after 

combustion, the entire blowdown and refill process may be 

described by a single nondimensional governing equation.

exit exit exit 4 u A

dt

d t = −ρ′ ′ ′ ′

ρ′ (3)

Here u ≡ u at3 ′ , A ≡ A Aref ′ , t′ ≡ tat3Aref V , a is the speed 

of sound, V is the combustor volume and Aref is an arbitrary 

reference area. The other states Pt4

′ and Tt4

′ are found from 

Equation (2) and standard isentropic relations and the equation 

of state.

4

4 4 4 4 ; t

t t CV t t

P P P T

ρ′

′ ′ = ′ ρ′ ′ = γ (4)

Furthermore, under the ideal nozzle assumption, all quantities 

on the right hand side of Equation (3) are also functions of 

Pt4

′ , Tt4′ , and a specified exit pressure (Ref. 14). As such, 

Equation (3) could be numerically integrated to obtain the 

temporal distribution of the combustor total conditions.

However, for the purposes of this work, such integration is not 

necessary. The interest here is not in the variation of properties 

with time per se, it is in the variation of properties with each 

infinitesimal unit of mass that leaves the combustor and enters 

the turbine. Such a unit of mass is simply 

dmexit ′ = ρ′

exituexit ′ Aexit ′ dτ which, from Equation (3), is dρt4 − ′ .

Thus, the distribution of properties with respect to exiting 

mass may be obtained by simply dividing the nondimensional 

mass associated with each cycle into a nearly arbitrary number 

of segments between the values of 1 and 0 (e.g., 1.0, 0.9, 0.8, 

etc.) and using Equation (4) to calculate the relevant states.

Such a distribution is shown in Figure 1. The heat addition 

parameter is q0

′ = 4.3 . This corresponds to a typical 

hydrocarbon fuel with a heating value of 18,266 Btu/lbm, and 

an air/fuel ratio of 43.7. The value of γ is 1.33. The station 3 

conditions correspond to a compressor pressure ratio of 10, 

operating at sea level static conditions, with an adiabatic 
NASA/TM—2012-217443 3

efficiency of 0.9. The inflection point shown at 

mexit ≡ mexit minitial ′ = 0.55 corresponds to the moment when 

the combustor has completed the blowdown process and has 

commenced refilling. It can be calculated as

(− γ ) ′ = − ′ 1 mexit_refill 1 PCV (5)

This distribution contains all the information needed to 

assess turbine work extraction and to perform the averages to 

be described.

2.2 Turbine Efficiency Map and Work 

Extraction

Turbine performance is often described using maps

(Ref. 15) which describe corrected flow rate and efficiency as 

functions of pressure ratio and corrected speed. For this study

however, only the efficiency is needed. The curves comprising 

an efficiency map are unique to a given turbine design.

However, owing to the physics of work extraction, they all 

have a similar shape. That is, they all possess a maximum at or 

near the design pressure ratio and speed, and fall off on either 

side of the design point. This similarity makes it possible to 

scale a turbine map such that it can represent (albeit crudely) 

nearly any class of engine (Ref. 16). An example of such a 

scaled map is shown in Figure 2. It is the one used in the 

present study. The figure shows efficiency as a function of 

scaled pressure ratio for various corrected speeds. Two 

families of curves are shown. One represents a turbine which 

is sensitive to operation away from the design point. The other 

is less sensitive. The range of turbine sensitivity shown is 

arbitrary and intended only to examine if sensitivity impacts 

unsteady work extraction.

Examining Figure 1, and considering the shape of the 

Figure 2 curves, it is clear that maximum work extraction will 

occur with a turbine designed for a pressure ratio PRt=Pt4/Pt5

= 4 5 / Pt Pt ′ ′ somewhere between 10 and 28 (with e.g., Pt5

′ =

1.05). However, there is very little known data on turbine 

efficiency for pressure ratios significantly above the design 

point. As such, the region of Figure 2 to the left of the peak 

value can be traced to the literature (Ref. 15), while the region 

to the right is somewhat speculative (except that it decreases).

Nevertheless, it suffices for the purposes of this study.

The impact of turbine design pressure ratio is illustrated in

Figure 3. Here, the ‘less sensitive’ turbine efficiency of 

Figure 2 at design corrected speed is replotted. Also shown are 

efficiencies calculated throughout the Figure 1 example cycle 

using the present map with different PRt_design values. It is 

assumed here and throughout the paper that the turbine 

physical speed is fixed over the course of a given PGC cycle. 

It can be seen that, although all curves reach peak efficiency

somewhere during the cycle, the values of PRt_design yielding 

higher mass-averaged efficiency are preferred.

Figure 1.—Model PGC process showing distributions (jump 

and decay) of pressure and temperature as functions of 

exiting mass in a combustor segment over the course of one 

cycle. The upstream compressor pressure ratio is 10. The 

equivalence ratio is 0.325.

Figure 2.—Scaled turbine efficiency map with notional curves 

representing a design point sensitive turbine and one that is 

less sensitive.

Figure 3.—Trajectory of the Figure 1 example cyclic combustor 

conditions on the scaled turbine efficiency map for different 

design pressure ratios.

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

P't4 and T't4

mexit/minitial

Pressure

Temperature

refill

blowdown

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.2 1.2 2.2

Turbine Efficiency, 

ηt

(PRt-1)/(PRt_design-1)

less sensitive

sensitive

N′

c=130%

N′

c=100%

N′

c=80%

0.6

0.7

0.8

0.9

1.0

0.2 1.2 2.2

Turbine Efficiency, 

ηt

(PRt-1)/(PRt_design-1)

less sensitive

PRt_design=10

PRt_design=28

PRt_design=19

N′

c=100%

PRt_design=10

PRt_design=28

PRt_design=19 
NASA/TM—2012-217443 4

Figure 4.—Distribution of work extraction and cycle massaverages for various values of PRt_design using the Figure 1

combustor conditions and the scaled turbine map.

In order to calculate cycle mass-averaged work extraction, 

the turbine map just described is used with the combustor 

output typified by Figure 1, and an imposed (assumed 

constant) turbine exit total pressure, Pt5, in the following 

integral*

.

( )

( 4 4 _ design _ design )

1

0

4 exit

initial 3

, , , ,

1 1 1

t t t t c t

t t t

g t

t t

f PR T T N PR

T PR dm m R T

W W

η = ′ ′

′































 = η ′ − γ

γ − ′ ≡ ∫ 















γ

γ− −

(6)

The integrand and integral of Equation (6) are shown in 

Figure 4 using the Figure 1 data and the same three PRt_design

values used in Figure 3. Also shown are the integrand and 

integral using a uniform turbine efficiency equal to the peak 

value of Figure 2. The impact of PRt_design is clearly seen. The 

maximum mass-averaged work extraction is obtained with a 

value of PRt_design=19.

This example case of turbine optimization illustrates two 

early conclusions which may be drawn from the current study.

First, maximum turbine work extraction for a PGC system 

requires a turbine designed for a considerably higher pressure 

ratio than for a conventional constant pressure combustor at 

similar operating conditions. Second, the work extracted will 

always be less than if the turbine is assumed to operate at the 

peak efficiency throughout the cycle.

 *

The imposition of constant Pt5 is done in order to close the problem; 

however, it cannot be formally justified. It is known that 

nonuniformity is reduced across a single turbine stage (Ref. 1 and 7). 

Here it is thus assumed that the turbine contains enough stages to 

provide complete uniformity. Practically, some nonuniformity will 

undoubtedly remain.

2.3 Averaging Procedures

The method of evaluating turbine work extraction just 

described represents the authors’ preferred low-order estimate 

for evaluating the benefits of PGC in the gas turbine 

environment. Furthermore, it is noted that the process of 

subjecting a turbine map to the output of a PGC model can be 

done with any PGC model or simulation. The idealized, 

lumped volume approach used here is convenient, but not 

necessary.

However, there remains a desire to assign a single pressure 

to a PGC device which represents availability; an equivalent

pressure which can be used as a figure-of-merit for different 

PGC designs. Among other applications, such a performance 

metric could be utilized in existing steady-state based cycle 

decks in order to evaluate mission benefits. As such, the 

objective of this and the following section of the paper are to 

describe various means of evaluating the average exit state 

which can then be applied to a turbine map and compared, on 

a work extraction basis, to the preferred method.

2.3.1 Mass-Averaged Pressure and Temperature ( P,T )

This is perhaps the most straightforward averaging 

technique. For the PGC model used here, the mass-averaged 

temperature is already known via the conservation of energy.

( ) Tt4 1 1 q0 ′ = + γ − ′ (7)

The mass-averaged pressure is found, like all massaverages, as

∫ ′ = ′ ′

1

0

Pt4 Pt4dmexit (8)

Unfortunately, while this averaging technique is simple it 

can be shown that the entropy associated with it is less than 

the mass-averaged entropy.

























′

′ ′ =

























′

′ < ′ =

























′

′ ′ =





 





γ−

γ 



 





γ−

γ 



 





γ−

γ

∫ CV

CV

t

t

t

t

PT P

T dm

P

T S

P

T S

1 1 1

ln ln ln

1

0

exit

4

4 4

4

4

4 _ (9)

As such, it would be expected that the average pressure 

calculated is optimistically high, as is the turbine work 

extraction calculated by inserting these averages in the steady 

form of Equation (6). 

2.3.2 Mass-Averaged Entropy and Temperature ( S ,T )

In this type of averaging the exit state is assigned the massaveraged temperature and entropy shown in Equations (7) and 

(9), respectively. The associated pressure may be written as 

follows:

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

Turbine Work,W'

t

mexit/minitial

PRt_design=10

PRt_design=28

PRt_design=19

eta_t=0.90

Color coded dashed lines indicate mass averages

ηt=0.90 

PRt_design=10

PRt_design=28

PRt_design=19 
NASA/TM—2012-217443 5





 





γ−

γ

















′

′ ′ = ′

1

4

4 _

CV

t

t ST CV T

T P P (10)

This type of averaging may be performed and interpreted 

another way that is conceptually useful. If the combustor 

exhaust is allowed to expand ideally with the imposed turbine 

exit pressure interpreted as a static value, then the turbine exit 

temperature becomes fixed and the following equations hold.

( ) 1

5

4

4 _ 5

1

5 5 ; 4

γ−

γ

γ

γ−

′

















′

′ ′ = ′ ′ = ′

s

t

t ST s S s s T

T T P e P P (11)

( )

2 ( 1) 4 5 2

5

γ −

′ − ′ = ′ u i Tt Ts (12)

Equation 12 represents the ideal mass-averaged kinetic 

energy that can be developed by the combustor flow. Thus, 

equivalent total pressure calculated via either Equation (10) or 

(11) may be interpreted as the total pressure which, when 

ideally expanded from the mass-averaged total temperature to 

the turbine exit static pressure, produces the mass-averaged 

kinetic energy. This is a useful interpretation since it makes 

somewhat intuitive sense that availability for turbine work 

extraction might be closely related to availability for

developing kinetic energy. On the other hand, the fact that no 

additional entropy is generated in the averaging process (i.e., 

there is no penalty in making the flow uniform) suggests that 

this averaging method too, may prove optimistic.

2.3.3 Ideal Specific Thrust and Mass-Averaged 

Temperature ( C,T )

Assuming again that each exiting mass unit ideally expands 

with the imposed turbine exit pressure interpreted as a static 

value, the ideal specific thrust, or mass-averaged spouting 

velocity may be calculated as follows.

( ) 5

1

0

exit 2

1

4 5

2

1

5 1

2

u i Tt

′ −Ts′ dm′ ≡ C′ 















γ − ′ = ∫ (13)

Comparing Equations (12) and (13), it is noted that 2

5

2 u5i u i ′ ≠ ′ . The significance of this fundamental inequality is 

discussed in References 17 and 18. Equation 13 is then used to 

obtain an equivalent or mixed static pressure.

2

5 _ 4 5 2

1 Ts CT Tt u i  ′



 



 γ − ′ = ′ − (14)

Figure 5.—Equivalent turbine work extraction error and 

combustor pressure calculated with the various averaging 

methods described. The Figure 1 combustor conditions were 

used.

From Equation (14), the equivalent total pressure is then 

readily obtained as

1

5 _

4

4 _ 5

γ−

γ

















′

′ ′ = ′

s CT

t

t CT s T

T P P (15)

Using an equation similar to Equation (9), it can be seen 

that this form of averaging yields a state with greater entropy 

than the mass-averaged exit value. That is to say, it reflects a 

loss in availability.

The three averaging methods just described were exercised 

on the example cycle illustrated in Figure 1. The resulting 

equivalent total pressures and temperatures were used with the 

optimized turbine map (PRt_design = 19) to determine efficiency, 

and turbine work via the steady form of Equation (6). This 

equivalent work extraction can be compared to the massaveraged work extraction shown in Figure 4. Figure 5 shows 

such comparisons along with the equivalent combustor 

pressures calculated using each of the methods. For this single 

case study, the spouting velocity or specific thrust averaging 

method produces the best representation of turbine availability 

in the sense that the subsequent turbine work calculated most 

closely matches the mass-averaged turbine work.

2.4 Statistical Study

In order to capture the possible influences of various 

operating and design conditions on the averaging methods 

discussed above, a statistical study was conducted. This study 

used the same thermodynamic cycle, or lumped volume, 

analysis methods described above, as well as the same 

parametric turbine map. The range of design and operating 

conditions are provided in Table I. The design point was 

assumed to be at an equivalence ratio of 0.4. All lower 

equivalence ratios were assumed to be off-design points.

1.25

1.30

1.35

1.40

1.45

1.50

1.55

0%

2%

4%

6%

C, T s, T P,T

P't4_Equ

W't_Eff/W't_Act - 1, %

Work

P'4

C ,T S ,T P,T

Pt4

′

Wt

′

,%1 W

W Equ

t

_t − ′

′
NASA/TM—2012-217443 6

TABLE I.—RANGE OF CONDITIONS EXAMINED

Parameter Value

Mach ........................................................................0.0, 0.5

Altitude ................................................................Sea Level

CPR......................................................................10, 20, 30

NPR................................................ 1.01 (M=0 only), 1.5, 2

φdesign ...............................................................................0.4

φoff-design .............................................................0.3, 0.2, 0.1

Turbine map................................... Sensitive, less sensitive

Similar to the above analysis, the turbine design point was 

optimized for each on-design condition to provide the 

maximum turbine work extraction. A total of 135 points were 

assessed. Several different averaging techniques were studied 

besides those presented thus far; however, they are not 

reported here as they did not provide superior correlation to 

the turbine work available. 

A comparison between the combustor pressure ratio 

computed using the three averaging techniques and the 

pressure ratio corresponding to a steady pressure providing the 

same turbine work at the mass-averaged combustor exit 

entropy is provided in Figure 6. Similar to the findings of 

Section 2.3, the specific thrust, or spouting velocity, average 

provides the best correlation, with an average error of 

2 percent. The mass-averaged entropy and temperature 

provides the next best correlation with an average error 

4 percent. The mass-averaged pressure and temperature 

provides the poorest correlation with an average error of 

nearly 7 percent. It should also be noted that the latter two 

methods consistently predict combustor pressure ratios larger 

than the average pressure ratio for the work available. That is 

to say, they consistently over-predict availability.

Finally, comparisons of mass-averaged turbine work 

extraction to equivalent pressure-based steady work extraction

using the three averaging techniques are shown in Figure 7.

Once again, the specific thrust, or mass-averaged spouting 

velocity, provides the most accurate correlation for work 

available. Shown in the figure are the maximum, minimum, 

average and standard deviation of the error in predicted work 

available for each of the previously described averaging 

techniques compared to the actual mass-averaged work 

computed. Similar to the combustor pressure ratio predictions, 

the mass-averaged pressure and temperature technique 

consistently over-predicts the work extraction. Although not 

shown, it is generally found that the disparities are larger for 

the sensitive turbine map than for the less sensitive map.

3.0 Detonative Combustion

In order to cursorily examine the impact of other PGC 

modes on the mass-averaged turbine work and averaging 

approaches presented, a detonative combustor cycle was 

examined. The combustor inlet conditions, air/fuel ratio, 

fuel heating value, gas constant, and ratio of specific heats 

were identical to the Figure 1 example. Simulation was done 

Figure 6.—Combustor pressure ratio corresponding to a 

steady pressure providing the mass-averaged turbine work 

versus equivalent combustor pressure ratio using the 

averaging techniques described.

Figure 7.—Turbine work extraction error calculated with the 

various averaging methods described. The Table 1 engine 

conditions were used.

using a high resolution, reactive, quasi-one-dimensional,

computational fluid dynamics (CFD) code. Details of the code 

can be found in the literature and will not be presented here in 

order to maintain focus on the results (Refs. 9 and 12). The 

simulated combustor section has uniform cross sectional area

over 93 percent of its length. The remaining length smoothly 

transitions to an exit throat with cross sectional area 

30 percent that of the uniform cross section region. Combined

with an imposed exit static pressure set to 50 percent of the 

inlet total, choked exit flow is assured throughout the cycle 

and the average fill Mach number is held to approximately 

0.26.

The complete cycle is conveyed in the plots of Figure 8. 

which show color contours of nondimensional pressure, 

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1.0 1.2 1.4 1.6 1.8

Equivalent P't4

P'

t4 to produce mass averaged work

Intg dT

Mass Avg P

Mass Avg T

Avg C0,I & T

PT

ST

CT

matched

5.4%

7.4%

11.9%

-0.7% -0.4%

1.2%

2.0% 1.6%

4.0%

1.5% 2.1% 2.4%

-5%

0%

5%

10%

15%

CT sT PT

Error in Predicted Work Available

max

min

avg

std dev

C ,T S ,T P,T

,%1 W

W Equ

t

_t − ′

′
NASA/TM—2012-217443 7

Figure 8.—Contours of pressure, temperature, Mach number and reactant fraction for 

the sample detonative combustor over the course of one limit cycle.

Figure 9.—Detonative PGC process showing distributions of 

exit plane pressure and temperature as functions of exiting 

mass in a combustor segment over the course of one cycle. 

The upstream compressor pressure ratio is 10. The 

equivalence ratio is 0.325.

temperature, Mach number and reactant fraction (mass 

fraction of detonable mixture) inside the combustor over the 

course of one limit cycle. Shown next to each contour are the 

maximum and minimum values of the variable found in the x-t 

space.

The exit flow from this simulation can be integrated to 

produce a similar detonative distribution to Figure 1. This is 

shown in Figure 9. Note that the range of variation is

considerably greater in Figure 9 than in Figure 1. This

variation has a substantial impact on the work extraction 

possible with the detonative combustor, as it results in a much 

broader spanning of the Figure 2 efficiency map. This is 

shown in Figure 10 which, like Figure 3, displays the 

efficiency of each mass unit exiting the detonative combustor 

over the course of the cycle. The design pressure ratio yielding 

the greatest work extraction from the less sensitive turbine 

map is PRt_design=17. It can be seen that the variation noted in 

Figure 9 yields efficiency as low as 0.39. When this 

distribution is integrated in Equation (6) it is found that, even 

though the detonative and constant volume combustion 

examples have exactly the same chemical energy added, and 

even though detonative combustion can be shown to produce 

less entropy than constant volume combustion, the optimized 

detonation-driven turbine of this example only extracts 

92 percent of the work as that of the constant volume-driven 

turbine.

These results indicate the challenges inherent in coupling a 

detonative combustor with a turbine in order to realize the 

combustor’s thermodynamic benefits. The broad temperature 

and pressure swings of the detonative combustion create 

structural and performance challenges for the turbine. The 

technical community is actively working the challenges 

through combined combustor and turbine research efforts

(Refs. 19 and 20). Furthermore, the detonative result above is 

specific to the extreme off-design behavior of the turbine 

efficiency map used which, as stated earlier, is speculative.

Data-based extended turbine maps are needed before any firm 

conclusions can be drawn.

t′

P′ T′ Mach 

Reactant

Fraction

x/L x/L x/L x/L

reflection

detonation

High

Low

fill

exhaust

7.04

0.50

3.49

0.95

1.96

-0.35

1.00

0.00

0

2

4

6

8

10

12

14

16

18

20

0.0 0.2 0.4 0.6 0.8 1.0

P't4 and T't4

mexit/minitial

Pressure

Temperature

refill

blowdown
NASA/TM—2012-217443 8

Figure 10.—Turbine efficiency for each mass unit exiting the 

detonative combustor described by Figure 8.

Figure 11.—Equivalent turbine work extraction error and 

combustor pressure calculated with the various averaging 

methods described. The Figure 1 combustor conditions were 

used with the Figure 8 detonative combustor results.

The averaging methods presented can be applied to the 

detonative combustor example and the equivalent steady work 

extraction can be compared to the mass-averaged work 

extraction as was done for Figure 5. This is shown in 

Figure 11. As with previous results, it is seen that the specific 

thrust or spouting velocity averaging provides the best match 

to ‘actual’ mass-averaged work extraction. However, even this 

relatively good match shows a much larger error than in any of 

the constant volume combustor results. Furthermore, and 

again although it is not shown, the error is even larger for the 

sensitive turbine map.

4.0 Discussion

Referring to Figure 5, Figure 7, and Figure 11, and 

assuming that the integral of Equation (6) represents the 

‘actual’ mass-averaged turbine work extraction, it is not 

surprising that the errors between actual work, and work 

calculated via the stagnation states resulting from the three 

averaging methods come out as they do. All three methods 

utilize the mass-averaged total temperature. This is 

appropriate, as it is a mathematically rigorous measure of the 

chemical energy added to the working fluid. It is therefore the 

various averaged pressures that warrant closer examination.

Figure 3 and Figure 4, as well as Equation (6), demonstrate 

that there is a performance penalty (i.e., the turbine is ondesign for only a small fraction of the entire mass processed) 

associated with nonuniform flows. The best pressure 

averaging method should produce a loss in availability that is 

consistent with this penalty.

As mentioned earlier, using the mass-averaged total 

pressure is straightforward and arguably intuitive. However, as 

shown in Equation (9), it results in a state which violates the 

second law of thermodynamics. Averaging procedures may be 

interpreted as mixing calculations where a nonuniform flow is 

forced (presumably through dissipative processes such as 

shearing) to a uniform state. Dissipative forces produce 

entropy and the mixing calculation should reflect this. Since 

mass averaging of the total pressure does not, it would be 

expected to produce the most unrealistically high combustor 

pressure ratios which in turn produce unrealistically high 

work extraction estimates. Figure 5, Figure 6, Figure 7, and 

Figure 11 show this to be precisely the case.

Calculating total pressure by preserving mass-averaged total 

temperature and entropy (Eqs. (10) to (12)) would also be 

expected to produce optimistically high combustor pressure 

ratios. Again, interpreted as a mixing calculation, this method 

exacts no entropy producing penalty on the nonuniform flow.

However, the method does preserve the ideal mass-averaged 

kinetic energy that a given PGC can produce. For purely thrust 

producing devices, exhaust kinetic energy is directly 

proportional to cycle thermal efficiency. It stands to reason 

then that this thermal efficiency should be related to 

availability for turbine work extraction. As such, the result that 

the mass-averaged total temperature and entropy averaging 

method produces smaller errors than the mass-averaged total 

temperature and pressure method is expected.

The averaging method preserving mass-averaged 

temperature and ideal specific thrust or mass-averaged 

spouting velocity produces the smallest work extraction errors.

As noted, this is the only averaging procedure that produces 

entropy and thus at least cursorily accounts for the work 

extraction penalty illustrated in Figure 3 and Figure 4.

Furthermore, there is a more direct physical rationale for the 

reduced error. It can be shown that the work extraction from 

an ideal Pelton-style impulse turbine operating at constant 

speed is written as follows (Ref. 18).

Wt

′ = 2U′(u5′i −U′) (16)

Practical turbines with multiple stages, degrees of reaction, 

significant axial flow components, and aerodynamic losses 

have a much more complex work formula; however, the 

fundamentally proportional relationship between spouting 

velocity and work extraction remains. Thus, an averaging 

method based on preserving the ideal spouting velocity is 

expected to, and does, yield turbine work extraction that 

closely matches the actual value.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0

Turbine Efficiency, 

ηt

(PRt-1)/(PRt_design-1)

less sensitive

PRt_design=17

N′

c=100%

PRt_design=17 

1.10

1.30

1.50

1.70

1.90

2.10

5%

10%

15%

20%

25%

30%

C, T s, T P,T

P't4_Equ

W't_Eff/W't_Act - 1, %

Work

P'4

C ,T S ,T P,T

Pt4

′

Wt

′

,%1 W

W Equ

t

_t − ′

′
NASA/TM—2012-217443 9

That being said, it is noted that in the case of the detonative 

combustor, even the ‘best’ spouting velocity averaging 

technique substantially over-predicts availability. The 

technique produces entropy scaled with the level of flow 

nonuniformity. Apparently however, it does not produce 

enough entropy to match the turbine performance penalty 

illustrated in Figure 10. On the other hand, it is quite possible 

that this performance penalty arising from an essentially

notional efficiency map is unrealistically harsh.

The detonative combustor results bring to light another 

topic worthy of passing discussion. It is presumed in the 

dynamic work extraction analysis that the PGC is directly 

coupled to the turbine. There are PGC concepts in the 

literature which place a mixing device between the PGC and 

the turbine21. Such devices serve to smooth the flow, though, 

as with the mixing calculations, at the fluidic cost of 

producing entropy. Nevertheless, it is entirely possible that for 

PGC devices with large variations in the flow effluent, the lost 

availability from mixing (or partial mixing) followed by 

turbine work extraction from a smoother flowfield could yield 

more work extraction than direct coupling to a turbine. This is 

particularly possible if the off-design turbine performance is 

as sensitive as the example shown. Significant investigations 

are needed to optimize this trade space.

5.0 Conclusions

A methodology was presented for calculating the massaveraged work extracted by a turbine exposed to the 

fundamentally nonuniform flowfield presented by a pressure 

gain combustor (PGC). The method is relatively simple, yet 

provides realistic values for so-called ‘actual’ work extraction.

This actual work extraction was compared to steady work 

extraction calculations based on turbine inlet states obtained 

by averaging simulated PGC effluent using several different 

averaging procedures. The objective was to determine which, 

if any averaging process provides a representative effective 

steady state in terms of availability for turbine work 

extraction. The resulting state, a combustor total exit pressure 

and temperature, may be used as a convenient performance 

metric for any PGC system without having to actually 

dynamically couple it to a turbine.

An averaging procedure was found which preserves massaveraged temperature and ideal specific thrust, or spouting 

velocity, and provides the best match to the actual work 

extraction. In a statistical exercise using an idealized constant 

volume PGC simulation, the work extraction predicted using 

this averaging technique was well matched to the actual work 

extraction over a wide range of combustor operating points, 

with an overall standard deviation of 1.5 percent. When a 

single operating point comparison was made for a detonative 

combustor, the specific thrust averaging technique still 

provided the best match; however, the error in work prediction 

was approximately 8 percent too high.

Though not a direct objective of the study, it was also 

found that the design turbine pressure ratio for maximum 

mass-averaged work extraction is significantly higher 

(approximately 70 percent) than that for a turbine fed by a 

constant pressure combustor with similar inlet conditions and 

equivalence ratio. This finding, while preliminary, has 

substantial implications when considering the actual design of 

PGC gas turbines. 

In closing, it is reiterated that this study presents a relatively 

simple, fundamental method to estimate the potential work 

available from a pressure gain combustor coupled with a 

turbine. More accurate performance predictions are available 

from higher fidelity dynamic analyses. Practical combustor 

and turbine component designs, as well as accurate 

performance predictions, will require such high fidelity 

renderings of the true fluid dynamics.

NASA/TM—2012-217443 11

Appendix—Nomenclature

A cross sectional area

C spouting velocity

CPR compressor pressure ratio, Pt3/P0

∆hv fuel lower heating value

Nc corrected turbine wheel speed

NPR nozzle pressure ratio, Pt5/P0

P pressure

PRt turbine total pressure ratio

L combustor length

Rg real gas constant

S entropy

T temperature

U turbine wheel velocity

V combustor volume

Wt turbine work

a speed of sound

a/f air to fuel ratio (by mass)

m mass

q0 mixture heating value

t time

u velocity

x axial distance

ηt turbine efficiency

φ equivalence ratio, ( )

(a f )

a f stoichiometric

γ ratio of specific heats

ρ density

τ nondimensional time

Subscripts

CV Constant Volume

Equ equivalent

design at the design point

exit combustor exit plane

i ideal

initial point in combustor cycle when filling with 

reactant is complete

off-design off of the design point

ref reference

s static

t total

0 ambient

3 compressor outlet

4 turbine inlet (combustor exit)

5 turbine exit

Superscripts

′ nondimensional

− mass-averaged
NASA/TM—2012-217443 12

References

1. Caldwell, N., and Gutmark, E., “Performance Analysis 

of a Hybrid Pulse Detonation Combustor / Gas Turbine 

System,” AIAA 2008-4880, July 2008.

2. Perkins, H. D., et al., “An Assessment of Pulse 

Detonation Engine Performance Estimation Methods 

Based On Experimental Results,” AIAA 2005-3831, July 

2005.

3. Paxson, D. E., Perkins, H. D., “Thermal Load 

Considerations for Detonative Combustion-Based Gas 

Turbine Engines,” AIAA-2004-3396, July 2004.

4. Rasheed, A., Tangirala, V. E., Vandervort, C. L., Dean, 

A. J., Haubert, C., “Interactions of a Pulsed Detonation 

Engine with a 2D Blade Cascade,” AIAA 2004-1207, 

January 2004.

5. Petters, D. and Felder, J., “Engine System Performance 

of Pulse Detonation Concepts Using the NPSS 

Program,” AIAA 2002-3910. July 2002.

6. Smith, C. F., Snyder, P. H., Emmerson, C. W., Nalim, 

M. R., “Impacts of the Constant Volume Combustor on a 

Supersonic Turbofan Engine,” AIAA-2002-3916, July 

2002.

7. Rouser, K. P., King, P. I., Schauer, F. R., Sondergaard 

R., Hoke, J. L., “Parametric Study of Unsteady Turbine 

Performance Driven by a Pulse Detonation Combustor,” 

AIAA 2010-6536, July 2010.

8. Kojima, et al., “Design Study of Turbine for Pulse 

Detonation Combustor,” AIAA 2007-5081, July 2007.

9. Paxson, D. E., Brophy, C., Bruening, G., “Performance 

Evaluation of a Pulse Detonation Combustion Based 

Propulsion System Using Multiple Methods,” JANNAF 

Journal of Propulsion and Energetics, Vol. 3, No. 1, 

May 2010, pp. 44-55.

10. Kaemming, T., Dyer, R., “The Thermodynamic and 

Fluid Dynamic Functions of a Pulsed Detonation Engine 

Nozzle,” AIAA-2004-3916, July 2004.

11. Cumpsty, N. A., Horlock, J. H., “Averaging Nonuniform 

Flow for a Purpose,” Journal of Turbomachinery, V. 

128, No. 1, pp. 120-129, January 2006.

12. Paxson, D.E., “Performance Evaluation Method for Ideal 

Airbreathing Pulse Detonation Engines,” Journal of 

Propulsion and Power, Vol. 20, No. 5, pp. 945-950, 

2004.

13. Suresh, A., Hofer, D.C., Tangirala, V.E. “Turbine 

Efficiency for Unsteady, Periodic Flows,” AIAA 2009-

0504, January 2009.

14. Anderson, J. D., Modern Compressible Flow, McGrawHill, N.Y., 1982, pp. 11-38.

15. Glassman, A. J., ed., Turbine Design and Application, 

NASA SP-290, 1994.

16. Jones, S.M., “An Introduction to Thermodynamic 

Performance Analysis of Aircraft Gas Turbine Engine 

Cycles Using the Numerical Propulsion System 

Simulation,” NASA/TM—2007-214690, March 2007.

17. Foa, J. V., Elements of Flight Propulsion, Wiley, New 

York, 1959, pp. 274-283.

18. Paxson, D. E., “A Simplified Model for Detonation 

Based Pressure-Gain Combustors,” AIAA 2010-6717, 

July 2010.

19. Kojima, et al., “Design Study of Turbine for Pulse 

Detonation Combustor,” AIAA 2007-5081, July, 2007.

20. Suresh, A., Hofer, D., Tangirala, V., “Turbine Efficiency 

for Unsteady Periodic Flows,” AIAA 2009-504, January

2009.

21. Paxson, D. E., Dougherty, K. T., “Ejector Enhanced 

Pulsejet Based Pressure Gain Combustors: An Old Idea 

With a New Twist,” AIAA 2005-4216, August 2005.

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

A methodology is described whereby the work extracted by a turbine exposed to the fundamentally nonuniform flowfield from a 

representative pressure gain combustor (PGC) may be assessed. The method uses an idealized constant volume cycle, often referred to as an 

Atkinson or Humphrey cycle, to model the PGC. Output from this model is used as input to a scalable turbine efficiency function (i.e., a 

map), which in turn allows for the calculation of useful work throughout the cycle. Integration over the entire cycle yields mass-averaged 

work extraction. The unsteady turbine work extraction is compared to steady work extraction calculations based on various averaging 

techniques for characterizing the combustor exit pressure and temperature. It is found that averages associated with momentum flux (as 

opposed to entropy or kinetic energy) provide the best match. This result suggests that momentum-based averaging is the most appropriate 

figure-of-merit to use as a PGC performance metric. Using the mass-averaged work extraction methodology, it is also found that the design 

turbine pressure ratio for maximum work extraction is significantly higher than that for a turbine fed by a constant pressure combustor with 

similar inlet conditions and equivalence ratio. Limited results are presented whereby the constant volume cycle is replaced by output from a 

detonation-based PGC simulation. The results in terms of averaging techniques and design pressure ratio are similar. 

15. SUBJECT TERMS

Gas turbine engine; Combustion 

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